Let now plane or spur teeth be struck upon the circles Pp and Ii, such as would cause them to drive one another as they would be driven by their mutual contact; that is, let these circles Pp and Ii be taken as the pitch circles of such teeth, and let the teeth be described, by any of the methods before explained, so that they may drive one another correctly. Let, moreover, their pitches be such, that there may be placed as many such teeth on the circumference Pp as there are to be teeth upon the bevil wheel HP, and as many on Ii as upon the wheel HI.

Having struck upon a flexible surface as many of the first teeth as are necessary to constitute a pattern, apply it to the conical surface DLMP, and trace off the teeth from it upon that surface, and proceed in the same manner with the surface DIXY.

Take DH equal to the proposed lengths of the teeth, draw ef through H perpendicular to AD, and terminate the wheels at their lesser extremities by concave surfaces HGml and HKxy described in the same way as the convex surfaces which form their greater extremities. Proceed, moreover, in the construction of pattern teeth precisely in the same way in respect to those surfaces as the others; and trace out the teeth from these patterns on the lesser extremities as on the greater, taking care that any two similar points in the teeth traced upon the greater and lesser extremities shall lie in the same straight line passing through A. The pattern teeth thus traced upon the two extremities of the wheels are the extreme boundaries or edges of the teeth to be placed upon them, and are a sufficient guide to the workman in cutting them.

two arcs, Pp, Ii, for the pitch circles of the teeth, and to set off the pitches upon them of the same lengths as the pitches upon the circles DP and DI, which last are determined by the numbers of teeth required to be cut upon the wheels respectively.

230. To prove that teeth thus constructed will work truly

with one another.

It is evident that if two exceedingly thin wheels had been taken in a plane perpendicular to AD (fig. p. 312.) passing through the point D, and having their centres in E and F; and if teeth had been cut upon these wheels according to the pattern above described, then would these wheels have worked truly with one another, and the ratio of their angular velocities have been inversely that of ED to FD, or (by similar triangles) inversely that of ND to OD; which is the ratio required to be given to the angular velocities of the bevil wheels.

Now it is evident that that portion of each of the conical surfaces DPML and DIXY which is at any instant passing through the line LY is at that instant revolving in the plane perpendicular to AD which passes through the point D, the one surface revolving in that plane about the centre E and the other about the centre F; those portions of the teeth of the bevil wheels which lie in these two conical surfaces will therefore drive one another truly, at the instant when they are passing through the line LY, if they be cut of the forms which they must have had to drive one another truly (and with the required ratio of their angular velocities) had they acted entirely in the above-mentioned plane perpendicular to AD and round the centres E and F. Now this is precisely the form in which they have been cut. Those portions of the bevil teeth which lie in the conical surfaces DPML and DIXY will therefore drive one another truly at the instant when they pass through the line LY; and therefore they will drive one another truly through an exceedingly small distance on either side of that line. Now it is only through an exceedingly small distance on either side of that line that any two given teeth remain in contact with one another. Thus then it follows, that those portions of the teeth which lie in the conical surfaces DM and DX work truly with one another.

Now conceive the faces of the teeth to be intersected by an infinity of conical surfaces parallel and similar to DM and DX; precisely in the same way it may be shown that those portions of the teeth which lie in each of this infinite number of conical surfaces work truly with one another; whence it follows that the whole surfaces of the teeth, constructed as above, work truly together.

231. THE MODULUS OF A SYSTEM OF TWO CONICAL OR BEVIL WHeels.

Let the pressures P, and P, be applied to the conical wheels represented in the accompanying figure at perpendicular distances a, and a, from their axes CB and CG; let the length AF of their teeth be represented by b; let the distance of any point in this line from F be represented by x, and conceive it to be divided into an exceedingly great number of equal parts, each represented by Ar. Through each of these points of division imagine planes to be drawn

perpendicular to the axes CB and CG of the wheels, dividing the whole of each wheel into elements or laminæ of equal thickness; and let the pressures P, and P, be conceived to be equally distributed to these laminæ. The pressure thus dis

2

P2 represent

the two pressures thus applied to the extreme lamina AH and AK of the wheels, and let them be in equilibrium when thus applied to those sections separately and independently of the rest; then if R represent the pressure sustained along that narrow portion of the surface of contact of the teeth of the wheels which is included within these laminæ, and if R, and R, represent the resolved parts of the pressure R in the directions of the planes AH and AK of these laminæ, the pressures p1 and R1 applied to the circle AH are pressures in equilibrium, as also the pressures p2 and R, applied to the circle AK. If, therefore, we represent as before (Art. 216.) by m, and m, the perpendiculars from B and G upon the directions of R, and R2, and by L, and L2, the distances between the feet of the perpendiculars a,, m, and a,, m, we have (equation 236, 237), neglecting the weights of the wheels,

P1

P1 and P2 representing the radii of the axes of the two wheels, and, and, the corresponding limiting angles of resistance. Let γι and 2 represent the inclinations of the direction of R to the planes of AH and AK respectively; then

R1 = R cos. 71, R2=R cos. 71⁄2•

Now it has been shown in the preceding article, that the action of that part of the surface of contact of the teeth which is included in each of the lamina AH, AK, is identical with the action of teeth of the same form and pitch upon two cylindrical wheels AD and AL of the same small thickness, situated in a plane EAD perpendicular to AC, and having their centres in the intersections, b and g, with that plane of

the axes CB and CG produced. The reciprocal pressure R of the teeth of the element has therefore its direction in the plane EAD; and if its direction coincided with the line of centres DL of the two circles EA and AD, then would its inclinations to the planes of AH and AK be represented by DAH and LAK, or by ACB and ACG.

The direction of R is however, in every case, inclined to the line of centres at a certain angle, which has been shown (Art. 216.) to be represented in every position of the teeth, after the point of contact has passed the line of centres by (+); where represents the inclination to AL of the line A, which is drawn from the point of contact A of the pitch. circles to the point of contact of the teeth, and where represents the limiting angle of resistance between the surfaces of the teeth. To determine the inclination y, of RA to the plane of the circle AH, its inclination RAD to the line of centres being thus represented by (9+), and the inclination. of the plane AD, in which it acts, to the plane AH being DAH, which is equal to ACB, let this last angle be represented by ; and let Aa in the accompanying figure represent the intersection of the planes AD and AH; Aard representing a portion of the former plane and Aach of the latter. Let moreover Ar represent the direction of the pressure R in the former plane, and let Ad and Ah be portions of the lines AD and AH of the preceding figure. Draw re perpendicular to the plane Aach, and rd and ch parallel to Aa, and join dh; then rAc represents the inclination, of the direction of R to the plane AD, dAr represents the inclination (+6) of AR to AD, and dAh represents the inclination, of the planes AD and AH to one another. Also, since Aa is perpendicular to the plane Ahd, therefore dr is perpendicular to that plane,

.. rc=Arsin. y1 = Ad sec. (+7) sin. y,•

Also hd Ad sin. 1, but rc=hd,

... Ad sec. (9+) sin.y1 = Ad sin. ;

.. sin. 7, cos. (+) sin. .